Homogenizing media containing a highly conductive honeycomb substructure

TL;DR

This paper develops a homogenization approach for heat conduction in a 3D medium with a highly conductive honeycomb structure, revealing the asymptotic temperature behavior as the honeycomb's volume shrinks.

Contribution

It introduces a novel homogenization method for media with thin, highly conductive honeycomb layers, accounting for their vanishing volume and thickness effects.

Findings

Derived the governing system for temperature asymptotics.

Analyzed the influence of layer thickness on heat conduction.

Provided a mathematical framework for complex honeycomb structures.

Abstract

The present paper deals with the homogenization of the heat conduction which takes place in a binary three-dimensional medium consisting of an ambiental phase having conductivity of unity order and a rectangular honeycomb structure formed by a set of thin layers crossing orthogonally and periodically. We consider the case when the conductivity of the thin layers is in inverse proportion to the vanishing volume of the rectangular honeycomb structure. We find the system that governs the asymptotic behaviour of the temperature distribution of this binary medium. The dependence with respect to the thicknesses of the layers is also emphasized. We use an energetic method associated to a natural control-zone of the vanishing domain.